What does integrability of finite-gap or soliton potentials mean?

@article{Brezhnev2008WhatDI,
  title={What does integrability of finite-gap or soliton potentials mean?},
  author={Yurii V. Brezhnev},
  journal={Philosophical transactions. Series A, Mathematical, physical, and engineering sciences},
  year={2008},
  volume={366 1867},
  pages={923-45}
}
In the example of the Schrödinger/KdV equation, we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finite-dimensional Hamiltonian systems (stationary KdV equations). Three key objects in this field-new explicit Psi-function, trace formula and the Jacobi problem-provide a complete solution. The Theta-function language is derivable from these… CONTINUE READING

References

Publications referenced by this paper.
Showing 1-10 of 64 references

Inverse Sturm–Liouville problems. Zeist, The Netherlands: VSP

  • B. M. Levitan
  • 1987
Highly Influential
4 Excerpts

1977a The integration of non-linear equations by methods of algebraic geometry

  • I. M. Krichever
  • Funct. Anal. Appl. 11,
  • 1977
Highly Influential
5 Excerpts

1977b Methods of algebraic geometry in the theory of non-linear equations

  • I. M. Krichever
  • Russ. Math. Surv
  • 1977
Highly Influential
4 Excerpts

Methods of algebraic geometry in the theory of non - linear equations

  • I. M. Krichever
  • Russian Math . Surveys .
  • 1977
Highly Influential
4 Excerpts

The integration of non - linear equations by methods of algebraic geometry

  • I. M. Krichever
  • Functional Anal . Appl .
  • 1977
Highly Influential
4 Excerpts

Abelian functions and solitons

  • V. B. Matveev
  • Preprint no
  • 1976
Highly Influential
12 Excerpts

Mémoire sur l ’ intégration d ’ une classe d ’ équations différentielles du second ordre en quantités finies explicites

  • J. Liouville
  • Liouville ’ s J . IV
  • 1839
Highly Influential
4 Excerpts

Solitons and the inverse scattering transform

  • M. J. Ablowitz, H. Segur
  • 1981
Highly Influential
4 Excerpts

Finite band linear differential operators and Abelian varieties

  • B. A. Dubrovin
  • Russ. Math. Surv
  • 1976
Highly Influential
8 Excerpts

Asymptotic behaviour of the resolvent of Sturm–Liouville equations and the algebra of the Korteweg–de Vries equations

  • I. M. Gel’fand, L. A. Dikii
  • Russ. Math. Surv
  • 1975
Highly Influential
4 Excerpts

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